# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

16,270
questions

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### intersection cohomology when the resolution is not semi-small

When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection ...

**35**

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**7**answers

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### Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" ...

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**2**answers

654 views

### Building elliptic curves into a family

Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$ has rank r. When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with ...

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660 views

### Corank 4 hypersurface singularities

A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \...

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**1**answer

406 views

### Is there a way to check if a relative Hilbert Scheme is reduced?

More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...

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### cohomology groups of tensor product of sheaves

If L and M are two local systems on a space X, what can we say about the cohomology groups $H^i(X,L\otimes M)$ in terms of the cohomology of L and M? For example, can we determine their dimensions. ...

**17**

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**4**answers

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### Why do automorphism groups of algebraic varieties have natural algebraic group structure?

I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure.
But if the automorphism group of a variety has algebraic group structure, how do I know the ...

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209 views

### Sheaf isomorphism.

Suppose you have a curve $C$ such that deg$K_C =0$ and $\Gamma(C,\Omega_C^1) \neq 0$. Does this automatically imply that $\vartheta_C \equiv \Omega_C^1$? My thought is yes, I've seen a proposition (...

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822 views

### Is every subgroup of an algebraic group a stabilizer for some action?

Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that ...

**10**

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946 views

### Fukaya categories of hyperkahler reductions: general request for information

I'd really like to hear any references or information people have about the Fukaya categories of hyperkahler reductions of vector spaces (for more informations on the varieties, see Nick Proudfoot's ...

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342 views

### finding the closure when blowing a variety at a singularity

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a node at $(0,0)$ (and at some other points) (it's not a homework question, just a ...

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**1**answer

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### Topological version of Bogomolov’s question

I'm quoting a question from p. 753 of Gromov's recent paper Singularities, Expanders and Topology of Maps:
"Does there exist, for every closed oriented $n$-manifold $X_0$, a closed oriented $n$-...

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**2**answers

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### Higher genus closed string B-model

The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which ...

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**2**answers

962 views

### Is the tangent space functor from PD formal groups to Lie algebras an equivalence?

The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...

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**4**answers

5k views

### Easiest way to determine the singular locus of projective variety & resolution of singularities

For an affine variety, I know how to compute the set of singular points by simply looking at the points where the Jacobian matrix for the set of defining equations has too small a rank.
But what is ...

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**2**answers

363 views

### Points on algebraic stacks

I'm a bit confused concerning a definition in Laumon--Moret-Bailly. Perhaps someone could shed some light on the following.
It concerns the definition of (closed) point in Chapter 5. More precisely, ...

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### Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...

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**1**answer

397 views

### Semiclassical explanation of “Structured” spaces [closed]

We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured ...

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**4**answers

2k views

### Finding divisors on a curve

What is the best way to find an actual divisor of an affine curve? I.E. if I am interested in finding a canonical divisor of a curve in two variables, is there a general way to go about it? Do I need ...

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**2**answers

375 views

### Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space

What are the representations of $U(n)$ that induce (see link text) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known ...

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488 views

### Weierstrass points on rigid-analytic surfaces

Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for ...

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**4**answers

4k views

### Definition of étale for rings

Let $A \to B$ be a ring extension.
What is the definition of $B/A$ étale ?
When $A$ is a field, do we get a nice characterization ?

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**2**answers

6k views

### Projective closure of affine curve

Is there a generalized method to find the projective closure of an affine curve? For example, I read that the projective closure of $y^2 = x^3−x+1$ in $\mathbb{P}^2$ is $y^2z = x^3−xz^2+z^3$.
If I ...

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**4**answers

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### Near Trivial Quiver Varieties

So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup:
I've been looking at the simplest case that didn't look ...

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votes

**2**answers

310 views

### A technical question about derivations of sheaves on group schemes

Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O_G$ the structural sheaf of $G$.
Let $D_e : O_{G,e} \to k$ a derivation.
I would ...

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**3**answers

704 views

### Line bundles trivial after extension of the base-field

Let k be a field and let X be scheme over k. Let K be a field extension of k and denote by $X_K$ the base-change of X to Spec K. Under what conditions is the canonical map of Picard groups $Pic(X)\to ...

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5k views

### What does “linearly disjoint” mean for abstract field extensions?

All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...

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**2**answers

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### Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...

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**3**answers

1k views

### Holomorphic and antiholomorphic forms of projective space

For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\...

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**3**answers

438 views

### Nature of Invertible Sheaves in which there are no global sections.

EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...

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### How can I really motivate the Zariski topology on a scheme?

First of all, I am aware of the questions about the Zariski topology asked here and I am also aware of the discussion at the Secret Blogging Seminar. But I could not find an answer to a question that ...

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**1**answer

309 views

### Systems of conics

It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...

**16**

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**5**answers

2k views

### Elliptic Curves over F_1?

Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...

**3**

votes

**4**answers

990 views

### Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...

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**2**answers

747 views

### a question about Gromov-Witten invariant

Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?

**4**

votes

**2**answers

212 views

### On the dimension of moduli space of pointed curves with fixed Weierstrass semigroup

Does anyone know any information on the question of the dimension of moduli space of pointed curves with fixed Weierstrass semigroup? Some conjecture?

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**4**answers

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### What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...

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votes

**3**answers

935 views

### Why a subvariety of a variety of general type is of general type

It seems a well-known fact that subvarieties of a variety of general type containing a general point are also of general type. This fact is an essential property used to prove some extension theorems ...

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**3**answers

2k views

### Elkies' supersingularity theorem in higher dimension

The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...

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votes

**4**answers

385 views

### Is tensoring with a module representable iff it is locally free of finite rank?

Motivation:
It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...

**0**

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**3**answers

909 views

### About the intersection of two vector bundles

I´m looking for information about the intersection of two vector bundles (principally trivial bundles, but no necessarily). I´m trying to make a picture (literally) of reflexive finite generated ...

**52**

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**11**answers

9k views

### Non-commutative algebraic geometry

Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which ...

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**1**answer

704 views

### Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces

I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally ...

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### Why is a variety of general type hyperbolic?

I heard people mentioned this in one sentence, but don't see the reason.
Why a (smooth) variety of general type, i.e. an algebraic variety X with K_X big, is hyperbolic, i.e. has no non-constant map ...

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**8**answers

9k views

### Why is it useful to study vector bundles?

I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...

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**4**answers

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### Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...

**33**

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**5**answers

2k views

### Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...

**21**

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**1**answer

786 views

### Do DG-algebras have any sensible notion of integral closure?

Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...

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**3**answers

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### What is “restriction of scalars” for a torus?

I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am ...

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**7**answers

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### Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. ...